Using this theorem, we show a recurrence criterion for regular Dirichlet forms.
Let \({\rm diag}=\lbrace (x,y)\in X\times X \mid x=y\rbrace \) .
We now suppose that \(({\cal E},{\cal F})\)  has no killing measure
in the Beurling-Deny representation
([1]}); that is,
there exist a symmetric form \(({\cal E}^{(c)},{\cal F}\cap C_0(X))\) 
with the strongly local property (see [1]} for definition) and
a symmetric positive Radon measure \(J({\rm d}x\,{\rm d}y)\)  on \(X\times X\setminus {\rm diag}\)  such that
\({\cal E}(u,u)={\cal E}^{(c)}(u,u)+\iint _{X\times X\setminus {\rm diag}}(u(x)-u(y))^2\,J({\rm d}x\,{\rm d}y), \quad u\in {\cal F}\cap C_0(X).\) 
